### Abstract

Simple linear models with additive stochastic forcing have been rather successful in explaining the observed spectrum of important climate variables. Motivated by this, the authors analyze the spectral character of such a general stochastic system of finite dimension. The spectral matrix is derived in the case that the spectrum is a linear combination of dynamical variables and their stochastic forcings. It is found that the most convenient basis for analysis is provided by the normal modes. In general the spectrum consists of two pieces. The first "diagonal" piece is a symmetric Lorentzian curve centered on the normal mode frequencies with breadth and strength determined by the normal mode dissipation. The second cross-spectrum piece derives usually from the coherency of the stochastic forcing of two different normal modes. The cross-spectrum is smaller in magnitude than the corresponding two diagonal pieces. This relative magnitude is controlled by the Wiener coherency, which is equal to the magnitude of the correlation of the stochastic forcings of different normal modes. This new analysis framework is studied in detail for the ENSO case for which a two-dimensional stochastically forced oscillator has been previously suggested as a minimal model. It is found that the observed spectrum is rather easily reproduced given appropriate dissipation. Further, it is found that the cross-spectrum results in a phase-dependent enhancement or suppression of frequencies smaller than the dominant ENSO frequency. This therefore provides a new mechanism for decadal ENSO variability. Since the cross-spectrum is phase dependent, the decadal variability generated has a distinctive spatial character. The significance of the cross-spectrum depends on the Wiener coherency, which in turn depends on the statistics of the stochastic forcing.

Original language | English (US) |
---|---|

Pages (from-to) | 13-25 |

Number of pages | 13 |

Journal | Journal of the Atmospheric Sciences |

Volume | 68 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2011 |

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### Keywords

- Decadal variability
- ENSO
- Spectral models/analysis/distribution
- Stochastic models

### ASJC Scopus subject areas

- Atmospheric Science

### Cite this

**Spectral analysis of multidimensional stochastic geophysical models with an application to decadal ENSO variability.** / Kleeman, Richard.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Spectral analysis of multidimensional stochastic geophysical models with an application to decadal ENSO variability

AU - Kleeman, Richard

PY - 2011/1

Y1 - 2011/1

N2 - Simple linear models with additive stochastic forcing have been rather successful in explaining the observed spectrum of important climate variables. Motivated by this, the authors analyze the spectral character of such a general stochastic system of finite dimension. The spectral matrix is derived in the case that the spectrum is a linear combination of dynamical variables and their stochastic forcings. It is found that the most convenient basis for analysis is provided by the normal modes. In general the spectrum consists of two pieces. The first "diagonal" piece is a symmetric Lorentzian curve centered on the normal mode frequencies with breadth and strength determined by the normal mode dissipation. The second cross-spectrum piece derives usually from the coherency of the stochastic forcing of two different normal modes. The cross-spectrum is smaller in magnitude than the corresponding two diagonal pieces. This relative magnitude is controlled by the Wiener coherency, which is equal to the magnitude of the correlation of the stochastic forcings of different normal modes. This new analysis framework is studied in detail for the ENSO case for which a two-dimensional stochastically forced oscillator has been previously suggested as a minimal model. It is found that the observed spectrum is rather easily reproduced given appropriate dissipation. Further, it is found that the cross-spectrum results in a phase-dependent enhancement or suppression of frequencies smaller than the dominant ENSO frequency. This therefore provides a new mechanism for decadal ENSO variability. Since the cross-spectrum is phase dependent, the decadal variability generated has a distinctive spatial character. The significance of the cross-spectrum depends on the Wiener coherency, which in turn depends on the statistics of the stochastic forcing.

AB - Simple linear models with additive stochastic forcing have been rather successful in explaining the observed spectrum of important climate variables. Motivated by this, the authors analyze the spectral character of such a general stochastic system of finite dimension. The spectral matrix is derived in the case that the spectrum is a linear combination of dynamical variables and their stochastic forcings. It is found that the most convenient basis for analysis is provided by the normal modes. In general the spectrum consists of two pieces. The first "diagonal" piece is a symmetric Lorentzian curve centered on the normal mode frequencies with breadth and strength determined by the normal mode dissipation. The second cross-spectrum piece derives usually from the coherency of the stochastic forcing of two different normal modes. The cross-spectrum is smaller in magnitude than the corresponding two diagonal pieces. This relative magnitude is controlled by the Wiener coherency, which is equal to the magnitude of the correlation of the stochastic forcings of different normal modes. This new analysis framework is studied in detail for the ENSO case for which a two-dimensional stochastically forced oscillator has been previously suggested as a minimal model. It is found that the observed spectrum is rather easily reproduced given appropriate dissipation. Further, it is found that the cross-spectrum results in a phase-dependent enhancement or suppression of frequencies smaller than the dominant ENSO frequency. This therefore provides a new mechanism for decadal ENSO variability. Since the cross-spectrum is phase dependent, the decadal variability generated has a distinctive spatial character. The significance of the cross-spectrum depends on the Wiener coherency, which in turn depends on the statistics of the stochastic forcing.

KW - Decadal variability

KW - ENSO

KW - Spectral models/analysis/distribution

KW - Stochastic models

UR - http://www.scopus.com/inward/record.url?scp=79251475575&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79251475575&partnerID=8YFLogxK

U2 - 10.1175/2010JAS3546.1

DO - 10.1175/2010JAS3546.1

M3 - Article

AN - SCOPUS:79251475575

VL - 68

SP - 13

EP - 25

JO - Journals of the Atmospheric Sciences

JF - Journals of the Atmospheric Sciences

SN - 0022-4928

IS - 1

ER -